# A Certified Model Reduction Approach for Robust Parameter Optimization with PDE Constraints

## TL;DR

This paper presents a novel certified model reduction method for robust parameter optimization constrained by PDEs, effectively handling uncertainty and reducing computational costs through adaptive surrogate modeling.

## Contribution

It introduces an adaptive model order reduction technique with certification for robust PDE-constrained optimization under uncertainty, improving efficiency over traditional methods.

## Key findings

- The proposed method effectively approximates worst-case scenarios in uncertain PDE models.
- Numerical results demonstrate significant computational savings and accuracy.
- The approach provides certified reduced order models for complex PDE-based optimization.

## Abstract

We investigate an optimization problem governed by an elliptic partial differential equation with uncertain parameters. We introduce a robust optimization framework that accounts for uncertain model parameters. The resulting non-linear optimization problem has a bi-level structure due to the min-max formulation. To approximate the worst-case in the optimization problem we propose linear and quadratic approximations. However, this approach still turns out to be very expensive, therefore we propose an adaptive model order reduction technique which avoids long offline stages and provides a certified reduced order surrogate model for the parametrized PDE which is then utilized in the numerical optimization. Numerical results are presented to validate the presented approach.